It is shown that in some generic cases the identity component of the group of leaf preserving diffeomorphisms (with not necessarily compact support) on a foliated open manifold is perfect. Next, it is proved that it is also bounded, i.e. bounded with respect to any bi-invariant metric. It follows that the group is uniformly perfect as well.
The notion of Cr,s and C∞,s-diffeomorphisms is introduced. It is shown that the identity component of the group of leaf preserving C∞,s-diffeomorphisms with compact supports is perfect. This result is a modification of the Mather and Epstein perfectness theorem.
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Let E be an ideal of L^0 over σ-finite measure space (Ω,Σ μ) and let (X, || X) be a real Banach space. Let E(X) be a subspace of the space L^0(X) of μ-equivalence classes of all strongly Σ-measurable functions f : Ω → X and consisting of all those f ε L^0(X), for which the scalar function [...] belongs to E. Let E be equipped with a Hausdorff locally convex-solid topology ξ and let ξ stand for the topology on E(X) associated with ξ. We examine the relationship between the properties of the space (E(X), ξ) and the properties of both the spaces (E, ξ) and (X, ||X). In particular, it is proved that E(X) (embedded in a natural way) is an order closed ideal of its bidual iff E is an order closed ideal of its bidual and X is reflexive. As an application, we obtain that E(X) is perfect iff E is perfect and X is reflexive.
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